https://file.aiquickdraw.com/mathbot/file/173024442965371ypnjkk.jpg

To solve this problem, let's break down the information given and formulate the exponential growth model.

### Step 1: Formulate the Exponential Growth Model
1. **Starting Quantity**: The initial number of bacteria is \( 1500 \).
2. **Doubling Time**: The bacteria population doubles every 3 hours.

The general form of an exponential growth model is:
\[
f(t) = A \cdot 2^{\frac{t}{d}}
\]
where:
- \( A \) is the initial quantity (1500 bacteria),
- \( d \) is the doubling time (3 hours),
- \( t \) is the time in hours.

Substituting \( A = 1500 \) and \( d = 3 \):
\[
f(t) = 1500 \cdot 2^{\frac{t}{3}}
\]

### Step 2: Predict the Bacteria Population After 2 Days
Since there are 24 hours in a day, 2 days is \( 2 \times 24 = 48 \) hours.

Substitute \( t = 48 \) into the model:
\[
f(48) = 1500 \cdot 2^{\frac{48}{3}} = 1500 \cdot 2^{16}
\]

Calculating \( 2^{16} \):
\[
2^{16} = 65536
\]

Now multiply by 1500:
\[
f(48) = 1500 \cdot 65536 = 98304000
\]

Rounding to the nearest hundred thousand, the answer is:
\[
f(48) \approx 98,300,000 \text{ bacteria}
\]

### Final Answer
- **Exponential model**: \( f(t) = 1500 \cdot 2^{\frac{t}{3}} \)
- **Population after 2 days**: Approximately 98,300,000 bacteria

Would you like further explanation on any of the steps?

### Related Questions:
1. How would the model change if the doubling time was different?
2. What happens to the population if we calculate for a longer period, like a week?
3. How is the exponential growth model affected by environmental constraints?
4. Can we use this model to calculate the half-life of bacteria under adverse conditions?
5. How does exponential growth differ from linear growth in biological systems?

#### Tip:
When dealing with exponential growth, always check the base growth factor (like doubling or tripling time) to understand how quickly the quantity scales.

A bacteria culture starts with 1,500 bacteria and doubles in size every 3 hours. Find an exponential model for the size of the culture as a function of time t in hours. Use the model to predict how many bacteria there will be after 2 days. (Round your answer to the nearest hundred thousand.)

Learn how to find an exponential growth model for a bacteria culture starting with 1,500 bacteria and doubling every 3 hours. Predict the bacteria population after 2 days using this model, rounded to the nearest hundred thousand. Suitable for high school math students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation